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version 1.20, 2006/04/07 11:40:20

version 1.21, 2006/05/31 12:54:14

Line 1161

<s><margin.target id="marg59"/><emph type="italics"/>Va&longs;a ina<lb/>nia aëre e&longs;&longs;e <lb/>plena, vari<lb/>is probatur <lb/>experimen<lb/>tis.<emph.end type="italics"/></s>

<s>Quòd &longs;i prædictorum Va&longs;orum inver&longs;orum aquis immer<lb/>&longs;orum <emph type="italics"/>fundum quis perforet, aqua quidem per os demer&longs;um in ip&longs;a in<lb/>gredietur, aër verò per foramen exibit,<emph.end type="italics"/> adeo manife&longs;tè, vt non &longs;o-

<pb xlink:href="051/01/051.jpg" pagenum="20"/>lùm manui admotæ &longs;en&longs;um inferat, &longs;ed etiam aures &longs;ono de<lb/>mulceat, & quidem harmonico, &longs;i foramini aptentur fi&longs;tulæ <lb/>&longs;ono Harmonico edendo aptæ, prout ip&longs;emet Hero fieri <lb/>jubet in multis, quas de&longs;cribit in citato Libello, hydraulicis at<lb/>que pnevmaticis Machinis. </s><s>Quod etiam quotidie experimur <lb/>in Hortis Romanis, Tu&longs;culanis, Tiburtinis, & in ip&longs;o etiam <lb/>P. Athana&longs;ij Kircheri mu&longs;eo. </s><s><emph type="italics"/>Non igitur <gap/>rbitrandum e&longs;t,<emph.end type="italics"/> infert <lb/>Hero, <emph type="italics"/>in his quæ&longs;unt, vacui naturam quandam coacervatam ip&longs;am per<lb/>&longs;e ip&longs;am exi&longs;tere &longs;ed&longs;ecundùm exiguas partes in aëre, & humido, alijsque <lb/>corporib<gap/> di&longs;&longs;eminatam,<emph.end type="italics"/> etiam in ip&longs;o adamante, duri&longs;&longs;imo licet, <lb/>den&longs;i&longs;&longs;<gap/>oque, <emph type="italics"/>quippe qui neque igniri, neque rumpi pote&longs;t, & percu&longs;<lb/>

<pb xlink:href="051/01/051.jpg" pagenum="20"/>lùm manui admotæ &longs;en&longs;um inferat, &longs;ed etiam aures &longs;ono de<lb/>mulceat, & quidem harmonico, &longs;i foramini aptentur fi&longs;tulæ <lb/>&longs;ono Harmonico edendo aptæ, prout ip&longs;emet Hero fieri <lb/>jubet in multis, quas de&longs;cribit in citato Libello, hydraulicis at<lb/>que pnevmaticis Machinis. </s><s>Quod etiam quotidie experimur <lb/>in Hortis Romanis, Tu&longs;culanis, Tiburtinis, & in ip&longs;o etiam <lb/>P. Athana&longs;ij Kircheri mu&longs;eo. </s><s><emph type="italics"/>Non igitur arbitrandum e&longs;t,<emph.end type="italics"/> infert <lb/>Hero, <emph type="italics"/>in his quæ&longs;unt, vacui naturam quandam coacervatam ip&longs;am per<lb/>&longs;e ip&longs;am exi&longs;tere &longs;ed &longs;ecundùm exiguas partes in aëre, & humido, alijsque <lb/>corporibus di&longs;&longs;eminatam,<emph.end type="italics"/> etiam in ip&longs;o adamante, duri&longs;&longs;imo licet, <lb/>den&longs;i&longs;&longs;imoque, <emph type="italics"/>quippe qui neque igniri, neque rumpi pote&longs;t, & percu&longs;<lb/>

<arrow.to.target n="marg60"/><lb/>&longs;us in incudes & malleos totus ing reditur<emph.end type="italics"/> (vt cum Pl<gap/>io putat Hero; <lb/>quod tamen, Mer&longs;enno te&longs;te, experientiæ repugnat, nam ictu <lb/>mallei in varias partes di&longs;&longs;ilit, & ita pote&longs;t iguiri, vt pereat. </s><s><emph type="italics"/>) Hoc <lb/><expan abbr="aut&etilde;">autem</expan> ip&longs;iaccidit,<emph.end type="italics"/> &longs;ubiungit Hero, <emph type="italics"/>non quòd vacuo careat, &longs;ed ob conti<lb/>nuatam den&longs;itatem.<emph.end type="italics"/> </s><s>Corpora porro illa (aëris præ&longs;ertim) quæ <lb/>partes &longs;uas habent inter &longs;e&longs;e cohærentes, <emph type="italics"/>non|tamen ex omni parte, <lb/>&longs;ed interiecta habent inter valla quædam vacua,<emph.end type="italics"/> (quæ vacuola appel<lb/>lat Mer&longs;ennus) bellè comparat Hero arenarum cumulo, inter <lb/>

<arrow.to.target n="marg60"/><lb/>&longs;us in incudes & malleos totus ingreditur<emph.end type="italics"/> (vt cum Plinio putat Hero; <lb/>quod tamen, Mer&longs;enno te&longs;te, experientiæ repugnat, nam ictu <lb/>mallei in varias partes di&longs;&longs;ilit, & ita pote&longs;t igniri, vt pereat.) </s><s><emph type="italics"/>) Hoc <lb/><expan abbr="aut&etilde;">autem</expan> ip&longs;iaccidit,<emph.end type="italics"/> &longs;ubiungit Hero, <emph type="italics"/>non quòd vacuo careat, &longs;ed ob conti<lb/>nuatam den&longs;itatem.<emph.end type="italics"/> </s><s>Corpora porro illa (aëris præ&longs;ertim) quæ <lb/>partes &longs;uas habent inter &longs;e&longs;e cohærentes, <emph type="italics"/>non|tamen ex omni parte, <lb/>&longs;ed interiecta habent inter valla quædam vacua,<emph.end type="italics"/> (quæ vacuola appel<lb/>lat Mer&longs;ennus) bellè comparat Hero arenarum cumulo, inter <lb/>

<arrow.to.target n="marg61"/><lb/>cuius grana multæ aëris particulæ intercedunt; aitque, arenæ par<lb/>ticulas particulis aëris &longs;imiles e&longs;&longs;e, <expan abbr="aër&etilde;">aërem</expan> verò, qui inter arenas in<lb/>terjicitur, &longs;imilem vacuolis intra aërem contentis; quæ quidem <lb/>vocuola tunc di&longs;inere putat in aëre (alijsque corporibus) cùm <lb/>accedente vi quapiam conden&longs;atur aër; redire verò, cùm facta <lb/>remi&longs;&longs;ione in pri&longs;tinum ordinem re&longs;tituitur aër ob naturalem <lb/>contentionem, quâ, velut arcus inflexus, aut &longs;icca &longs;pongia vi <lb/>pre&longs;&longs;a, molem ab ip&longs;o rerum conditu &longs;ibi tributam repetit, &longs;ta<lb/>

<arrow.to.target n="marg62"/><lb/>tim atque vis externa de&longs;init. </s><s><emph type="italics"/>At aëris corpora,<emph.end type="italics"/> inquit Hero, <emph type="italics"/>inter<lb/>&longs;e&longs;e quidem cohærent, non tamen ex omni parte, &longs;edinteriecta habent <lb/>intervalla quædam vacua, &longs;icut arena, quæ e&longs;t in littoribus. </s><s>Itaque a<lb/>nimo concipiendum e&longs;t, arenæ particulas corporibus aëris &longs;imiles e&longs;&longs;e, aë<lb/>rem vero, qui inter particulas arenæ interiicitur, &longs;imilem vacuis intra <lb/>aërem contentis. </s><s>Quamobrem vi quadam accedente äerem den&longs;ari con<lb/>tingit, & in vacuorum loca re&longs;idere, corporibus præter naturam inter<lb/>&longs;e&longs;e compre&longs;&longs;is: remi&longs;sione verò factârar&longs;us in cun dem ordinem re&longs;titui-<emph.end type="italics"/>

<pb xlink:href="051/01/052.jpg" pagenum="21"/><emph type="italics"/>tur, ob naturalem Corporum contentionem; quemadmodum & in cor<lb/>nuum ramentis, & in &longs;pongiis &longs;iccis, quæ&longs;i compre&longs;&longs;a remittantur, rur<lb/>&longs;us in cundem locum redeunt, eandem que accipiunt molem. </s><s>Similiter <lb/>&longs;i aliqua vi aëris particulæ à &longs;e in vicem di&longs;tractæ fuerint, & maior præ<lb/>ter naturam locus vacuus fiat, rur&longs;us ad &longs;e&longs;e recurrunt, per vacuum e<lb/>nim celerem corporum lationem fieri contingit, nullo ob&longs;tunte, aut re<lb/>pellente, quóusque corpora ad &longs;e&longs;e applicentur.<emph.end type="italics"/></s>

Line 1171

<s><margin.target id="marg61"/><emph type="italics"/>Vacuola He<lb/>ronis inter <lb/>aëris parti<lb/>culas de&longs;&longs;e<lb/>minata.<emph.end type="italics"/></s>

<s><margin.target id="marg62"/><emph type="italics"/>A<gap/>mterie <lb/>Ha habens <lb/>vacua <expan abbr="&longs;ec&utilde;-dum">&longs;ecun<lb/>dum</expan> Hero<lb/>nem compa<lb/><gap/>atur are<lb/>næ.<emph.end type="italics"/></s>

<s><margin.target id="marg62"/><emph type="italics"/>Aër interie<lb/>cta habens <lb/>vacua <expan abbr="&longs;ec&utilde;-dùm">&longs;ecun<lb/>dùm</expan> Hero<lb/>nem compa<lb/>ratur are<lb/>næ.<emph.end type="italics"/></s>

Line 1207

<s><margin.target id="marg68"/><emph type="italics"/>Sphæra con<lb/>cava, & aë<lb/>re plena, cur <lb/>plùs aeris <lb/>per inflatio<lb/>nem recipi<lb/>at <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> <lb/>Heronem.<emph.end type="italics"/></s>

<s>Quartò, <emph type="italics"/>con&longs;truatur Sphæra cra&longs;&longs;itudinem habens laminæ, vt nè <lb/>facilè rumpatur, continensque circiter cotylas octo, & vndique præclu<lb/>&longs;a;<emph.end type="italics"/> & per foramen in ip&longs;a factum immittamus <emph type="italics"/>Siphonem æreum, <lb/>hoc e&longs;t, tubum gracilem, non contingentem locum, qui puncto perfora<lb/>to &longs;ecundùm diametrum opponitur, ita vt aqua fluere po&longs;&longs;it; alia autem <lb/>ip&longs;ius pars extra &longs;phæram emineat circiter digitos tres, ip&longs;i&queacute; diligenter <lb/>adferruminetur, ambitu foraminis circa tubum &longs;tanno præclu&longs;o, vt cùm <lb/>volumus ore per &longs;iphonem inflare, &longs;piritus ex &longs;phærâ nullo modo excidat.<emph.end type="italics"/><lb/> </s><s>His factis,, <emph type="italics"/>in&longs;piciamus quæ contingunt. </s><s>Aëre enim in ip&longs;a exi&longs;tente<emph.end type="italics"/><lb/>(quemadmodum & in aliis va&longs;is quæ vacua appellantur) <emph type="italics"/>totum<lb/>que qui in ipsa locum replente, & per continuationem quandam ad eius <lb/>ambitum applicato, nullo denique loco, ut arbitrantur, penitus exi&longs;ten<lb/>te vacuo; neque aquam immittere po&longs;&longs;emus, neque alium aërem, non <lb/>excedente priori aëre qui in ipsa erat; & &longs;i multa vi immi&longs;&longs;ionem facia<lb/>mus, priùs di&longs;rumpetur vas, quàm aliquid &longs;u&longs;cipiat, cùm plenum &longs;it: ne<lb/>que enim aëris corpora in minorem magnitudinem contrahi po&longs;&longs;unt: <lb/>quare nece&longs;&longs;e erit, vt in &longs;eip&longs;is habeant quædam intervalla, in quæ com<lb/>pre&longs;&longs;a minorem obtineant molem; hoc autem veri&longs;imile non e&longs;t, nullo <lb/>penitus exi&longs;tente vacuo: & cùm corpora ad &longs;e&longs;e applicentur &longs;ecundùm <lb/>omnes &longs;uperficies, & &longs;imiliter ad va&longs;is ambitum, non po&longs;&longs;unt expul&longs;a lo<lb/>cum alicui facere, non exi&longs;tente vacuo aliquo, quamobrem nullo modo <lb/>in &longs;phæram immittetur aliquid eorum, quæ &longs;unt extra, ni&longs;i excedat ali<lb/>qua pars aëris priùs in ea contenti, &longs;iquidem totus locus con&longs;tipatus est, <lb/>& continuatus, vt arbitrantur. </s><s>At<emph.end type="italics"/> experientiâ con&longs;tat contrari<lb/>um: nam <emph type="italics"/>&longs;i quis &longs;iphonem ori admovens &longs;phæraminflet, multum im<lb/>mittit &longs;piritum, non excedente aëre, qui in ea e&longs;t. </s><s>Quod cùm ita &longs;em<lb/>per contingat; manife&longs;tò o&longs;tenditur, <expan abbr="contraction&etilde;">contractionem</expan> fieri <expan abbr="corpor&utilde;">corporum</expan> in &longs;phæra <lb/>exi&longs;tentium in vacua implicata: contractio autem præter naturam fit, <lb/>ob immi&longs;&longs;ionis violentiam. </s><s>Si igitur aliquis inflans, & ad ip&longs;um os ma<lb/>num apponens, &longs;tatim digito &longs;iphonem obturet; manebit omni tempore<emph.end type="italics"/>

<pb xlink:href="051/01/054.jpg" pagenum="23"/><emph type="italics"/>con&longs;tipatus aër in &longs;phæra: quod&longs;i quis aperiat, rur&longs;us extra erump<gap/>t cum <lb/><gap/>agno &longs;trepitu & clamore immi&longs;&longs;us aër, propterea quòd, vt propo&longs;uimus, <lb/>expellitur à dilatatione pr<gap/>exi&longs;tentis aëris impetu quodam factâ.<emph.end type="italics"/></s>

<pb xlink:href="051/01/054.jpg" pagenum="23"/><emph type="italics"/>con&longs;tipatus aër in &longs;phæra: quod&longs;i quis aperiat, rur&longs;us extra erumpet cum <lb/>magno &longs;trepitu & clamore immi&longs;&longs;us aër, propterea quòd, vt propo&longs;uimus, <lb/>expellitur à dilatatione præexi&longs;tentis aëris impetu quodam factâ.<emph.end type="italics"/></s>

<s>Quintò, <emph type="italics"/>&longs;i quis velit aërem, qui in Sphæra e&longs;t, per &longs;iphonem at</s>

Line 1663

<arrow.to.target n="marg120"/><lb/>ad v&longs;que propugnaculum, cui ruinam minantur, viam anfra<lb/>ctuo&longs;am, angu&longs;tamque, magneticæ pixidis ductu, uti inter alios <lb/>docet P. Athana&longs;ius Kircherus lib. 2. de Arte Magnet. par. 2. cap. 1. <lb/>Probl. 7. & nos in Pantometro Kircherianolib. 5. Ichnographi<lb/>co Probl. 8. </s><s>Sub loco deinde ruinæ de&longs;tinato Cameram con<lb/>&longs;truunt laxiorem, eamque implent pulveris tormentarij va&longs;is, <lb/>& o&longs;tium claudunt, relicto exiguo foramine, perque viam totam <lb/>ignis pabulum &longs;ternunt, & egre&longs;&longs;i ignem applicant; qui &longs;erpens <lb/>per viam accendit pulveris ma&longs;&longs;am omnem Camerâ conclu&longs;am, <lb/>itàque rarefacit, dilatatque, ut angu&longs;tiarum impatiens, & laxio<lb/>rem quærens locum, impo&longs;itam &longs;ibi molem di&longs;iiciat horribili <lb/>

<arrow.to.target n="marg121"/><lb/>fremitu, nec minori terrore, quàm damno. </s><s>Nihil unquam <lb/>magis &longs;imile terræ motui, qui & ip&longs;e rarefactionis e&longs;t effectus lu-

<pb xlink:href="051/01/093.jpg" pagenum="62"/>culenti&longs;&longs;imus, excogitatum fuit. </s><s>Eodem enim pror&longs;us modo <lb/>in terræ concu&longs;&longs;ione flamma ex aliqua &longs;ubterranea fornace, vel <lb/>interno calore, per venulam aliquam nitri aut &longs;ulphuris &longs;erpit, <lb/>donec perveniat ad locum, in quo maiorem huiu&longs;ce materiæ <lb/>copiam inveniat; quâ &longs;ubitò accen&longs;a & rarefactâ, dum maio<lb/>rem quærit locum, &longs;ibi impo&longs;itas concutit, & di&longs;jicit non rarò, <lb/>moles. </s><s>Sæpe etiam iam antea accen&longs;us in terræ vi&longs;ceribus ignis, <lb/>dum nova augetur aliunde attracta materia, montes &longs;ibi impo&longs;i<lb/>

<arrow.to.target n="marg122"/><lb/>tos concutit, finditque, vt non &longs;emel audivi, vidique in Ætna Si<lb/>ciliæ, duorum annorum accola olim, & nunc hoc ip&longs;o anno 1654. <lb/>quo hæc &longs;cribo, hîc Romæ, quam (raro exe n<gap/>o) concu&longs;&longs;it mo<lb/>tus ille ingens, qui tantam Soræ &longs;tragem intulit, & Romam hinc <lb/>inde Neapolim, omnes que interiacentes regiones tremefecit. </s>

<arrow.to.target n="marg122"/><lb/>tos concutit, finditque, vt non &longs;emel audivi, vidique in Ætna Si<lb/>ciliæ, duorum annorum accola olim, & nunc hoc ip&longs;o anno 1654. <lb/>quo hæc &longs;cribo, hîc Romæ, quam (raro exemplo) concu&longs;&longs;it mo<lb/>tus ille ingens, qui tantam Soræ &longs;tragem intulit, & Romam hinc <lb/>inde Neapolim, omnes que interiacentes regiones tremefecit. </s>

<s><margin.target id="marg119"/><emph type="italics"/>Rarefactio<lb/>nis vis in <lb/>bombardis.<emph.end type="italics"/></s>

Line 2357

<s><emph type="center"/>Annotatio I.<emph.end type="center"/></s>

<s><emph type="italics"/>OMninò nece&longs;&longs;arium e&longs;t, ut locus<emph.end type="italics"/> E <emph type="italics"/>&longs;it humilior, &longs;altem aliquan<lb/>tulùm, quàm locus fontis aut aquæ<emph.end type="italics"/> B, <emph type="italics"/>&longs;eu C, ut &longs;iphonis crus<emph.end type="italics"/> DE <lb/><emph type="italics"/>longius fieri po&longs;&longs;it quàm<emph.end type="italics"/> BD, <emph type="italics"/>aut<emph.end type="italics"/> DC. </s><s><emph type="italics"/>Quod &longs;i fieri non po&longs;&longs;it, &longs;pes <lb/>nulla e&longs;t ad effectum de&longs;ideratum con&longs;equendum. </s><s>Quare diligenter <lb/>priùs libellandum e&longs;t &longs;patium inter<emph.end type="italics"/> B <emph type="italics"/>&<emph.end type="italics"/> E, <emph type="italics"/>aut<emph.end type="italics"/> C <emph type="italics"/>&<emph.end type="italics"/> E. <emph type="italics"/>Qu&<gap/>grave; <lb/>autem humilior fuerit locus<emph.end type="italics"/> E <emph type="italics"/>quàm aqua<emph.end type="italics"/> B <emph type="italics"/>aut<emph.end type="italics"/> C, <emph type="italics"/>eòmcliorem res <lb/>&longs;ucce&longs;&longs;um habebit, & eò velociùs <gap/>qua effluet ex orificio<emph.end type="italics"/> E, <emph type="italics"/>vti ex di<lb/>ctis patet.<emph.end type="italics"/></s>

<s><emph type="italics"/>OMninò nece&longs;&longs;arium e&longs;t, ut locus<emph.end type="italics"/> E <emph type="italics"/>&longs;it humilior, &longs;altem aliquan<lb/>tulùm, quàm locus fontis aut aquæ<emph.end type="italics"/> B, <emph type="italics"/>&longs;eu C, ut &longs;iphonis crus<emph.end type="italics"/> DE <lb/><emph type="italics"/>longius fieri po&longs;&longs;it quàm<emph.end type="italics"/> BD, <emph type="italics"/>aut<emph.end type="italics"/> DC. </s><s><emph type="italics"/>Quod &longs;i fieri non po&longs;&longs;it, &longs;pes <lb/>nulla e&longs;t ad effectum de&longs;ideratum con&longs;equendum. </s><s>Quare diligenter <lb/>priùs libellandum e&longs;t &longs;patium inter<emph.end type="italics"/> B <emph type="italics"/>&<emph.end type="italics"/> E, <emph type="italics"/>aut<emph.end type="italics"/> C <emph type="italics"/>&<emph.end type="italics"/> E. <emph type="italics"/> Qu&<gap/>grave; <lb/>autem humilior fuerit locus<emph.end type="italics"/> E <emph type="italics"/>quàm aqua<emph.end type="italics"/> B <emph type="italics"/>aut<emph.end type="italics"/> C, <emph type="italics"/>eòmcliorem res <lb/>&longs;ucce&longs;&longs;um habebit, & eò velociùs aqua effluet ex orificio<emph.end type="italics"/> E, <emph type="italics"/>vti ex di<lb/>ctis patet.<emph.end type="italics"/></s>

<s><emph type="center"/>Annotatio II.<emph.end type="center"/></s>

Line 2373

<s><emph type="center"/>Siphone inver&longs;o aquam ex montis radice ad <lb/>ejus verticem elevare.<emph.end type="center"/></s>

<s>MIrabilis e&longs;t &longs;equens modus, & antiquis minimè cognit<gap/><lb/>

<s>MIrabilis e&longs;t &longs;equens modus, & antiquis minimè cognitur, <lb/>

<arrow.to.target n="marg186"/><lb/>imò ab omnibus pro impo&longs;&longs;ibili habitus, inquit Port<gap/><lb/>omnes enim putârunt, &longs;i in &longs;iphonis inver&longs;i collo &longs;eu curva<gap/><lb/>ra &longs;uprema foramen fieret, & vel minimum aëris ingrede<gap/><lb/>tur, fore ut &longs;iphonis aqua di&longs;continuaretur, & aqua divi&longs;a p<gap/><lb/>utrumque &longs;iphonis crus hinc atque inde delaberetur, totu&longs;q<gap/><lb/>&longs;iphon evacuaretur. </s><s>Contrarium mon&longs;trat &longs;equens modus, <lb/>quem præ&longs;cribit Joannes Bapti&longs;ta Porta lib. 2. Spirital. cap. 13. <lb/>& 14. in hunc &longs;en&longs;um. <lb/>

<arrow.to.target n="marg186"/><lb/>imò ab omnibus pro impo&longs;&longs;ibili habitus, inquit Porta: <lb/>omnes enim putârunt, &longs;i in &longs;iphonis inver&longs;i collo &longs;eu curvatu<lb/>ra &longs;uprema foramen fieret, & vel minimum aëris ingredere<lb/>tur, fore ut &longs;iphonis aqua di&longs;continuaretur, & aqua divi&longs;a per <lb/>utrumque &longs;iphonis crus hinc atque inde delaberetur, totu&longs;que <lb/>&longs;iphon evacuaretur. </s><s>Contrarium mon&longs;trat &longs;equens modus, <lb/>quem præ&longs;cribit Joannes Bapti&longs;ta Porta lib. 2. Spirital. cap. 13. <lb/>& 14. in hunc &longs;en&longs;um. <lb/>

<arrow.to.target n="marg187"/></s>

<s><margin.target id="marg186"/><emph type="italics"/>Vide præ <lb/><expan abbr="cedent&etilde;">cedentem</expan> <lb/>figuram <lb/>pag<emph.end type="italics"/> 99. <lb/><emph type="italics"/>vel<emph.end type="italics"/> 100.</s>

<s><margin.target id="marg187"/><emph type="italics"/>Aquam è <lb/>montis ra<lb/>dice ad ver<lb/>ticem ele<lb/>vare.<emph.end type="italics"/></s>

<s>Sit ut antea mons A, in eju&longs;que radice fons B, cujus <lb/>aqua &longs;it elevanda u&longs;que ad verticem D. </s><s>Fiat &longs;iphon inver&longs;um <lb/>inæqualium, ut antea, crurium BDE, cujus orificium B &longs;it <lb/>immer&longs;um fonti, orificium verò E vel liberum omninò &longs;it; <lb/>vel de&longs;inat in pi&longs;cinam &longs;ubjectam. </s><s>Habebit autem res tantò<lb/>meliorem &longs;ucce&longs;ium, quantò longius fuerit crus DE, quàm<lb/>alterum DB. </s><s>Fiat deinde infra collum &longs;iphonis, ubi D, re<lb/>ceptaculum aquarum H, undique clau&longs;um quàm diligenti&longs;<lb/>&longs;imè, &longs;uoque epi&longs;tomio I in&longs;tructum, ex quo depromi aqua <lb/>po&longs;&longs;it. </s><s>In hoc receptaculum derivetur ex collo &longs;eu curvatum <lb/>&longs;iphonis tubulus G, habens claviculam volubilem, &longs;itque tu<lb/>bulus arcti&longs;&longs;imè adferruminatus & &longs;iphonis collo, & recept<gap/><lb/>culi operculo priùs perforatis. </s><s>His factis, repleatur per infu<gap/><lb/>dibulum D &longs;iphon BDE, ob&longs;tructis priùs orificijs B & <gap/><lb/>eoque repleto obturetur foramen D, & aperiatur &longs;imul utru<gap/><lb/>que orificium B & E; dumque per tubum fluit aqua, & ex<gap/><lb/>erumpit; aperiatur modici&longs;&longs;imè, & momentaneâ qua&longs;i mo<gap/><lb/>la clavicula tubuli G; &longs;tatimque per dictum tubulum de&longs;ce<gap/><lb/>det modica aqua intra receptaculum H, a&longs;cendetque mo<gap/><lb/>cus aër intra &longs;iphonem, ibique aquæ mi&longs;tus delabetur unà cu<gap/><lb/>ip&longs;a per crus DE, & per os E erumpet. </s><s>Hoc &longs;i &longs;æpius fece-

<s>Sit ut antea mons A, in eju&longs;que radice fons B, cujus <lb/>aqua &longs;it elevanda u&longs;que ad verticem D. </s><s>Fiat &longs;iphon inver&longs;um <lb/>inæqualium, ut antea, crurium BDE, cujus orificium B &longs;it <lb/>immer&longs;um fonti, orificium verò E vel liberum omninò &longs;it; <lb/>vel de&longs;inat in pi&longs;cinam &longs;ubjectam. </s><s>Habebit autem res tantò<lb/>meliorem &longs;ucce&longs;ium, quantò longius fuerit crus DE, quàm<lb/>alterum DB. </s><s>Fiat deinde infra collum &longs;iphonis, ubi D, re<lb/>ceptaculum aquarum H, undique clau&longs;um quàm diligenti&longs;<lb/>&longs;imè, &longs;uoque epi&longs;tomio I in&longs;tructum, ex quo depromi aqua <lb/>po&longs;&longs;it. </s><s>In hoc receptaculum derivetur ex collo &longs;eu curvatum <lb/>&longs;iphonis tubulus G, habens claviculam volubilem, &longs;itque tu<lb/>bulus arcti&longs;&longs;imè adferruminatus & &longs;iphonis collo, & recept<gap/><lb/>culi operculo priùs perforatis. </s><s>His factis, repleatur per infun<lb/>dibulum D &longs;iphon BDE, ob&longs;tructis priùs orificijs B & E <lb/>eoque repleto obturetur foramen D, & aperiatur &longs;imul utrum<lb/>que orificium B & E; dumque per tubum fluit aqua, & ex E <lb/>erumpit; aperiatur modici&longs;&longs;imè, & momentaneâ qua&longs;i motu<lb/>la clavicula tubuli G; &longs;tatimque per dictum tubulum de&longs;cen<lb/>det modica aqua intra receptaculum H, a&longs;cendetque modi<lb/>cus aër intra &longs;iphonem, ibique aquæ mi&longs;tus delabetur unà cum <lb/>ip&longs;a per crus DE, & per os E erumpet. </s><s>Hoc &longs;i &longs;æpius fece-

<pb xlink:href="051/01/134.jpg" pagenum="103"/>ris, tandem replebitur receptaculum H, ut aqua per ejus epi<lb/>&longs;tomium I depromi po&longs;&longs;it magna copia. </s><s>Hæc Porta. </s>

<s><emph type="center"/>Annotatio.<emph.end type="center"/></s>

Line 2429

<s><margin.target id="marg193"/><emph type="italics"/>Solutio Au<lb/>ctoris ratio <lb/>num dubi<lb/>tandi circæ <lb/>&longs;iphonem in<lb/>terruptum.<emph.end type="italics"/></s>

<s>Dico igitur, cau&longs;am e&longs;&longs;e, quòd aquæ pars unaquæque <lb/>premitur aquâ &longs;upra ip&longs;am exi&longs;tente ad perpendiculum, &longs;i aqua <lb/>illa &longs;it de&longs;cendens in aliquo, (hoc e&longs;t, &longs;i nullum impedimen<lb/>tum extrin&longs;ecum habeat, quò minùs de&longs;cendere po&longs;&longs;it) pro<lb/>ut a&longs;&longs;eruimus &longs;uprà cap. 1. hujus Protheoriæ 4. Propriet. 4. ex <lb/>Archimede. </s><s>Illa igitur aqua, cujus perpendiculum e&longs;t majus, <lb/>&longs;eu longius, magis premitur, <expan abbr="majorem&qacute;ue">majoremque</expan> habet vim ad de<lb/>&longs;cendendum, &longs;i patet aditus ad de&longs;cendendum. </s><s>Quoniam <lb/>igitur aperto epi&longs;tomio E tubi EF, in I. Fig. & aperto ore <lb/>tubi recti DE, in II. Fig. patet aditus aquæ ad de&longs;cendendum; <lb/>de&longs;cendet illa nece&longs;&longs;ariò, innato pondere, & intra tubum, <lb/>propter partes continuatas, fundabit &longs;uum perpendiculum; & <lb/>nè vacuum detur, trahet &longs;ecum aërem, & coget aquam oppo<lb/>fitorum va&longs;orum a&longs;cendere, <expan abbr="&longs;uum&qacute;ue">&longs;uumque</expan> &longs;imiliter perpendiculum <lb/>fundare. </s><s>Nunquam tamen a&longs;cendere pote&longs;t aqua ad majus <lb/>&longs;patium in tubo BK, & crure AB, quàm de&longs;cenderit per tu<lb/>bum EF, & DE; quia &longs;i a&longs;cenderet ad majus &longs;patium, jam <lb/>perpendiculum aquæ a&longs;cendentis majus e&longs;&longs;et, quàm perpen<lb/>diculum aquæ de&longs;cendentis, ac proinde illa vinceret hanc, & <lb/>retraheret. </s><s>Itaque &longs;i vas C haberet in fundo foramen angu<lb/>itum &longs;ine tubo adnexo, nihil aquæ efflueret, quia nullum fun<lb/>dare po&longs;&longs;et perpendiculum; ideoque nihil a&longs;cenderet per tu-

<pb xlink:href="051/01/138.jpg" pagenum="107"/>bum BK. </s><s>Si idem vas C haberet foramen laxum, efflueret <lb/>quidem aqua, &longs;ed &longs;imul &longs;ubintraret aër, &longs;icque ce&longs;&longs;ante metu <lb/>vacui, aut di&longs;continuitatis corporum, nihil a&longs;cenderet per tu<lb/>bum KB. </s><s>Si tubus EF e&longs;&longs;et brevic<gap/>tubo BK, de&longs;cenderet aqua <lb/>u&longs;que ad orificium F, & ad æquale &longs;patium a&longs;cenderet per tu<lb/>bum BK; &longs;ed facta hac æqualitate, &longs;i&longs;teretur utrimque aqua, <lb/>propter perpendiculorum æqualitatem. </s><s>Si tubus EF e&longs;&longs;et <lb/>præcisè æqualis tubo BK; de&longs;cenderet &longs;imiliter aqua u&longs;que <lb/>ad o&longs;culum F, & ex parte altera a&longs;cenderet u&longs;que ad K, &longs;ed <lb/>neutra ulteriùs pergeret, propter cau&longs;am jam dictam de per<lb/>pendiculorum æqualitate. </s><s>Si denique tubus EF, e&longs;t vel pau<lb/>lulum longior tubo BK, de&longs;cendit aqua u&longs;que ad F, & cùm <lb/>nullum inveniat impedimentum quò minùs egrediatur, nec <lb/>detineatur ab altera parte à perpendiculo æquali; effluit inna<lb/>to pondere ex F, & ut vitetur vacuum, <expan abbr="corporum&qacute;ue">corporumque</expan> di&longs;con<lb/>tinuitas, &longs;ecum rapit & a&ecaron;rem va&longs;is KA, & aquam tubi BK, <lb/>va&longs;i&longs;que B. </s>

<pb xlink:href="051/01/138.jpg" pagenum="107"/>bum BK. </s><s>Si idem vas C haberet foramen laxum, efflueret <lb/>quidem aqua, &longs;ed &longs;imul &longs;ubintraret aër, &longs;icque ce&longs;&longs;ante metu <lb/>vacui, aut di&longs;continuitatis corporum, nihil a&longs;cenderet per tu<lb/>bum KB. </s><s>Si tubus EF e&longs;&longs;et brevior tubo BK, de&longs;cenderet aqua <lb/>u&longs;que ad orificium F, & ad æquale &longs;patium a&longs;cenderet per tu<lb/>bum BK; &longs;ed facta hac æqualitate, &longs;i&longs;teretur utrimque aqua, <lb/>propter perpendiculorum æqualitatem. </s><s>Si tubus EF e&longs;&longs;et <lb/>præcisè æqualis tubo BK; de&longs;cenderet &longs;imiliter aqua u&longs;que <lb/>ad o&longs;culum F, & ex parte altera a&longs;cenderet u&longs;que ad K, &longs;ed <lb/>neutra ulteriùs pergeret, propter cau&longs;am jam dictam de per<lb/>pendiculorum æqualitate. </s><s>Si denique tubus EF, e&longs;t vel pau<lb/>lulum longior tubo BK, de&longs;cendit aqua u&longs;que ad F, & cùm <lb/>nullum inveniat impedimentum quò minùs egrediatur, nec <lb/>detineatur ab altera parte à perpendiculo æquali; effluit inna<lb/>to pondere ex F, & ut vitetur vacuum, <expan abbr="corporum&qacute;ue">corporumque</expan> di&longs;con<lb/>tinuitas, &longs;ecum rapit & a&ecaron;rem va&longs;is KA, & aquam tubi BK, <lb/>va&longs;i&longs;que B. </s>

<s>Notandum tamen e&longs;t, quod jam &longs;æpius inculcavi, alti<lb/>tudinem perpendiculi aquæ tubi BK &longs;umendam e&longs;&longs;e à &longs;upre<lb/>ma aquæ va&longs;is B, &longs;uperficie, u&longs;que ad &longs;upremam &longs;uperficiem <lb/>aquæ eju&longs;dem tubi. </s>

Line 2494

<arrow.to.target n="marg202"/><lb/><emph type="italics"/>perpendiculariter, aut inclinati, aut pro&longs;trati horizontaliter. </s><s><lb/>Priores vocabimus verticales; medios, inclinatos; ultimos <lb/>horizontales.<emph.end type="italics"/></s>

<s><margin.target id="marg202"/>Tubi verti<lb/><gap/>ales.</s>

<s><margin.target id="marg202"/>Tubi verti<lb/>cales.</s>

<s><emph type="italics"/>His ita expo&longs;itis, nunc afferemus varia Phænomena, <lb/>id e&longs;t, apparentias &longs;eu ob&longs;ervationes circa fluxum aquæ per tu<lb/>bos verticales, cuiuscunque formæ & conditionis; ex quibus <lb/>deinde deducemus varia Problemata atque Theoremata; quo<lb/>rum multa applicari etiam poterunt fluxui aquæ per &longs;iphones, <lb/>de quibus cap. præcedente. </s><s>Agimus autem hoc Capite &longs;olùm <lb/>de fluxu aquæ per tubos ex foramine in ba&longs;i facto; de fluxu enim <lb/>ex foramine facto in latere, loquemur capite &longs;equenti.<emph.end type="italics"/></s>

Line 2535

<s>ALtitudo tuborum, tum hac, tum &longs;equentibus Propo&longs;itioni <lb/>

<arrow.to.target n="marg205"/><lb/>bus, &longs;umitur à lumine &longs;eu foramine, u&longs;que ad &longs;uperius orifi<lb/>cium, includendo etiam ip&longs;ius foraminis altitudinem &longs;eu cra&longs;<lb/>&longs;itiem, ab interiori ad exteriorem &longs;uperficiem ba&longs;is, &longs;i in ba&longs;i &longs;eu <lb/>fundo &longs;it foramen. </s>

<s><margin.target id="marg205"/><emph type="italics"/>Tuborum <lb/>altitudo v<gap/><lb/>de &longs;umatur<emph.end type="italics"/></s>

<s><margin.target id="marg205"/><emph type="italics"/>Tuborum <lb/>altitudo vn<lb/>de &longs;umatur<emph.end type="italics"/></s>

<s>Sint igitur, ut antea, tubi, AB, æquè al<gap/>i, & æqualium fo<lb/>raminum B, &longs;ive &longs;emper pleni, &longs;ive non &longs;emper pleni, cuius<lb/>cunque capacitatis, & figuræ. </s><s>Dico, per foramina illa effluere <lb/>eodem, vel æquali tempore, æqualem aquæ copiam. </s><s>Quoni<lb/>am enim, per præcedentem Propo&longs;itionem, &longs;upra foramem B <lb/>in omnibus æqualis aquæ copia æquali vi premit, nempe colu<lb/>mna aquea AB, cuius ba&longs;is e&longs;t foramen B in omnibus æquale, <lb/>ut &longs;upponitur; & altitudo e&longs;t perpendicularis AB, in omnibus <lb/>itidem æqualis; nece&longs;&longs;e e&longs;t, æqualem aquæ copiam eodem tem<lb/>pore decurrere ex omnibus: ubi enim omnia &longs;unt æqualia, ef<lb/>fectus &longs;unt æquales. </s>

<s>Sint igitur, ut antea, tubi, AB, æquè alti, & æqualium fo<lb/>raminum B, &longs;ive &longs;emper pleni, &longs;ive non &longs;emper pleni, cuius<lb/>cunque capacitatis, & figuræ. </s><s>Dico, per foramina illa effluere <lb/>eodem, vel æquali tempore, æqualem aquæ copiam. </s><s>Quoni<lb/>am enim, per præcedentem Propo&longs;itionem, &longs;upra foramem B <lb/>in omnibus æqualis aquæ copia æquali vi premit, nempe colu<lb/>mna aquea AB, cuius ba&longs;is e&longs;t foramen B in omnibus æquale, <lb/>ut &longs;upponitur; & altitudo e&longs;t perpendicularis AB, in omnibus <lb/>itidem æqualis; nece&longs;&longs;e e&longs;t, æqualem aquæ copiam eodem tem<lb/>pore decurrere ex omnibus: ubi enim omnia &longs;unt æqualia, ef<lb/>fectus &longs;unt æquales. </s>

<s><emph type="center"/>Corollarium I.<emph.end type="center"/><lb/>

Line 2611

<arrow.to.target n="marg209"/><lb/>ip&longs;e lineas, alij &longs;crupulos vocant. </s><s>Vncia, &longs;eu duodecima pedis pars ap<lb/>pellatur etiam pollex; ab aliquibus verò cum Mer&longs;enno digitus: rectiùs <lb/>tamen digitus e&longs;t decima &longs;exta pars pedis, vt con&longs;tat ex Vitruvio, Iulio <lb/>Frontino, Columella, & alijs apud Villalpandum tom.<emph.end type="italics"/> 3. <emph type="italics"/>Apparatus <lb/>Part.<emph.end type="italics"/> 2. <emph type="italics"/>lib.<emph.end type="italics"/> 3. <emph type="italics"/>cap.<emph.end type="italics"/> 16. <lb/>

<arrow.to.target n="marg210"/></s>

<s><margin.target id="marg208"/><emph type="italics"/>Pes dividi<lb/>tur in vnci<lb/><gap/><emph.end type="italics"/> 12. <emph type="italics"/>& di<lb/>gitos<emph.end type="italics"/> 16.</s>

<s><margin.target id="marg208"/><emph type="italics"/>Pes dividi<lb/>tur in vnci<lb/>as<emph.end type="italics"/> 12. <emph type="italics"/>& di<lb/>gitos<emph.end type="italics"/> 16.</s>

<s><margin.target id="marg209"/>Linea e&longs;t <lb/>duodecima <lb/>pars vncia.</s>

<s><margin.target id="marg210"/><emph type="italics"/>Minuti &longs;e<lb/><gap/>undi dura<lb/>tio, quanta <lb/>&longs;it.<emph.end type="italics"/></s>

<s><margin.target id="marg210"/><emph type="italics"/>Minuti &longs;e<lb/>cundi dura<lb/>tio, quanta <lb/>&longs;it.<emph.end type="italics"/></s>

<s><emph type="italics"/>Minutum &longs;ecundum e&longs;t &longs;exage&longs;ima pars unius minuti primi, &longs;eu<emph.end type="italics"/><lb/>3600. <emph type="italics"/>pars vnius horæ, & proximè re&longs;pondet lento arteriæ &longs;eu cordis <lb/>pul&longs;ui, ut Mer&longs;ennus a&longs;&longs;erit.<emph.end type="italics"/></s>

Line 2645

<s>RAtio &longs;eu proportio duplicata quæcunque, e&longs;t ratio quæ<lb/>

<arrow.to.target n="marg214"/><lb/>cunque &longs;implex &longs;emel repetita, &longs;eu bis continuè &longs;umpta:

<pb xlink:href="051/01/149.jpg" pagenum="118"/>quemadmodum ratio quæcunque triplicata, quadruplicata, &c. <lb/>e&longs;t ratio quæcunque &longs;implex bis, ter &c. repetita, &longs;eu ter, <lb/>quater continuè &longs;umpta. </s><s>Exemplum. </s><s>Inter 2 & 1 reperitur <lb/>ratio dupla; hæc ratio &longs;i &longs;emel repetatur, &longs;eu adhuc &longs;emel <lb/>accipiatur, hoc e&longs;t, &longs;i bis continuè &longs;umatur hoc modo, 4, <lb/>2, 1; erit inter 4 & 1 ratio &longs;eu proportio duplicata illius pro<lb/>portionis, quæ e&longs;t inter 2 & 1, quandoquidem inter 4 & 1 <lb/>reperitur ratio dupla &longs;emel repetita, &longs;eu bis continuè <lb/>&longs;umpta, hoc e&longs;t, duplicata, &longs;cilicet &longs;emel inter 4 & 2, & ite<lb/>rum inter 2 & 1. </s><s>Similiter inter 8 & 1 e&longs;t ratio triplicata il<lb/>lius, quæ e&longs;t inter 2 & 1, quia inter 8 & 1, intercedit ter <lb/>ratio dupla, nempe 8 ad 4, 4 ad 2, 2 ad 1. </s><s>Sic 16 ad 1 ha<lb/>bet rationem quadruplicatam, & 32 ad 1 rationem quintu<lb/>plicatam rationis illius, quam habet 2 ad 1. </s><s>Aliud exemplum. </s><s><lb/>Inter 6 ad 4 reperitur ratio &longs;e&longs;quialtera &longs;implex; hæc ratio <lb/>duplicatur, &longs;i adhuc &longs;emel repetatur, &longs;eu &longs;i bis continuè &longs;uma<lb/>tur, ut apparet in his numeris 9, 6, 4: nam quia ut 6 ad 4, <lb/>ita 9 ad 6; ideo inter 9 & 4 bis reperitur ratio &longs;e&longs;quialtera. </s><s><lb/>Si verò eadem ratio &longs;e&longs;quialtera bis repetatur, &longs;eu ter conti<lb/>nuè ponatur; erit inter extremos terminos ratio &longs;e&longs;qui altera <lb/>triplicata, ut apparet in his numeris, 13<gap/>, 9, 6, 4; quam pro<lb/>portionem ab&longs;que fractione habebis, &longs;i duplicaveris ho&longs;ce <lb/>numeros &longs;ic, 27, 18, 12, 8: nam ut 12 continet 8 &longs;emel cum <lb/>dimidio, ita 18 continet 12 &longs;emel cum dimidio, & 27 etiam <lb/>continet 18 &longs;emel cum dimidio. <lb/>

<pb xlink:href="051/01/149.jpg" pagenum="118"/>quemadmodum ratio quæcunque triplicata, quadruplicata, &c. <lb/>e&longs;t ratio quæcunque &longs;implex bis, ter &c. repetita, &longs;eu ter, <lb/>quater continuè &longs;umpta. </s><s>Exemplum. </s><s>Inter 2 & 1 reperitur <lb/>ratio dupla; hæc ratio &longs;i &longs;emel repetatur, &longs;eu adhuc &longs;emel <lb/>accipiatur, hoc e&longs;t, &longs;i bis continuè &longs;umatur hoc modo, 4, <lb/>2, 1; erit inter 4 & 1 ratio &longs;eu proportio duplicata illius pro<lb/>portionis, quæ e&longs;t inter 2 & 1, quandoquidem inter 4 & 1 <lb/>reperitur ratio dupla &longs;emel repetita, &longs;eu bis continuè <lb/>&longs;umpta, hoc e&longs;t, duplicata, &longs;cilicet &longs;emel inter 4 & 2, & ite<lb/>rum inter 2 & 1. </s><s>Similiter inter 8 & 1 e&longs;t ratio triplicata il<lb/>lius, quæ e&longs;t inter 2 & 1, quia inter 8 & 1, intercedit ter <lb/>ratio dupla, nempe 8 ad 4, 4 ad 2, 2 ad 1. </s><s>Sic 16 ad 1 ha<lb/>bet rationem quadruplicatam, & 32 ad 1 rationem quintu<lb/>plicatam rationis illius, quam habet 2 ad 1. </s><s>Aliud exemplum. </s><s><lb/>Inter 6 ad 4 reperitur ratio &longs;e&longs;quialtera &longs;implex; hæc ratio <lb/>duplicatur, &longs;i adhuc &longs;emel repetatur, &longs;eu &longs;i bis continuè &longs;uma<lb/>tur, ut apparet in his numeris 9, 6, 4: nam quia ut 6 ad 4, <lb/>ita 9 ad 6; ideo inter 9 & 4 bis reperitur ratio &longs;e&longs;quialtera. </s><s><lb/>Si verò eadem ratio &longs;e&longs;quialtera bis repetatur, &longs;eu ter conti<lb/>nuè ponatur; erit inter extremos terminos ratio &longs;e&longs;qui altera <lb/>triplicata, ut apparet in his numeris, 13 1/4, 9, 6, 4; quam pro<lb/>portionem ab&longs;que fractione habebis, &longs;i duplicaveris ho&longs;ce <lb/>numeros &longs;ic, 27, 18, 12, 8: nam ut 12 continet 8 &longs;emel cum <lb/>dimidio, ita 18 continet 12 &longs;emel cum dimidio, & 27 etiam <lb/>continet 18 &longs;emel cum dimidio. <lb/>

<arrow.to.target n="marg215"/></s>

<s><margin.target id="marg214"/><emph type="italics"/>Duplicat<gap/><lb/>proportio <lb/>quæ.<emph.end type="italics"/></s>

<s><margin.target id="marg214"/><emph type="italics"/>Duplicata <lb/>proportio <lb/>quæ.<emph.end type="italics"/></s>

<s><margin.target id="marg215"/><emph type="italics"/>Subduplica<lb/>ta proportio <lb/>quæ.<emph.end type="italics"/></s>

Line 2686

<s><emph type="center"/>Aqua naturali motu de&longs;cendens & effluens per tubos, <lb/>imitatur leges aliorum gravium naturali <lb/>motu de&longs;cendentium.<emph.end type="center"/></s>

<s>NOn uni mirabile vi&longs;um, inquit Mer&longs;ennus in Phænome<lb/>nis Hydraulicis Propo&longs;it. III. quòd non &longs;it eadem ratio <lb/>quantitatis aquæ fluentis ex tubo pedali, ad aquæ quantitatem <lb/>ex quadrupedali tubo &longs;alientis, po&longs;ito æquali foramine in utro<lb/>que, quæ e&longs;t 1 ad 4; cùm aqua quadrupedalis tubi quadruplo <lb/>magis, quàm pedalis premere videatur fundum, & aquam ex <lb/>lumine &longs;alientem. </s><s>Verùm mirari de&longs;ines, ubi noveris, aquam <lb/>eo &longs;olùm modo premere, vel ea duntaxat velocitate tubum <lb/>egredi, quâ moveretur, &longs;i ex eadem tubi altitudine cecidi&longs;&longs;et; <lb/>adeo ut &longs;it eadem i&longs;tius phænomeni ratio, quæ de&longs;cen&longs;us aquæ <lb/>per tubum; & eadem hujus de&longs;cen&longs;us, quæ de&longs;cen&longs;us aliorum <lb/><gap/>m; de quo proinde hîc agendum paulò accuratiùs, quan<lb/><gap/>dem aquæ motus per tubum, & ex tubo eruptio, &longs;equi<lb/><gap/>ges gravium motu naturali de&longs;cendentium. </s>

<s>NOn uni mirabile vi&longs;um, inquit Mer&longs;ennus in Phænome<lb/>nis Hydraulicis Propo&longs;it. III. quòd non &longs;it eadem ratio <lb/>quantitatis aquæ fluentis ex tubo pedali, ad aquæ quantitatem <lb/>ex quadrupedali tubo &longs;alientis, po&longs;ito æquali foramine in utro<lb/>que, quæ e&longs;t 1 ad 4; cùm aqua quadrupedalis tubi quadruplo <lb/>magis, quàm pedalis premere videatur fundum, & aquam ex <lb/>lumine &longs;alientem. </s><s>Verùm mirari de&longs;ines, ubi noveris, aquam <lb/>eo &longs;olùm modo premere, vel ea duntaxat velocitate tubum <lb/>egredi, quâ moveretur, &longs;i ex eadem tubi altitudine cecidi&longs;&longs;et; <lb/>adeo ut &longs;it eadem i&longs;tius phænomeni ratio, quæ de&longs;cen&longs;us aquæ <lb/>per tubum; & eadem hujus de&longs;cen&longs;us, quæ de&longs;cen&longs;us aliorum <lb/>gravium; de quo proinde hîc agendum paulò accuratiùs, quan<lb/>doquidem aquæ motus per tubum, & ex tubo eruptio, &longs;equi<lb/>tur leges gravium motu naturali de&longs;cendentium. </s>

<s><emph type="center"/>Leges gravium naturali motu de&longs;cendentium.<emph.end type="center"/></s>

<s>LEges porrò, quas gravia in de&longs;cen&longs;u naturali motu facto <lb/>

<arrow.to.target n="marg219"/><lb/>&longs;ervant, incredibili cura ac diligentia indagavit & ob&longs;er<lb/>vavit Bononiæ P. Joannes Bapti&longs;ta Ricciolus, unà cum alijs <lb/>è no&longs;tra Societate Patribus, ea&longs;que clari&longs;&longs;imè ac di&longs;tincti&longs;&longs;imè <lb/>

<arrow.to.target n="marg220"/><lb/>proponit in Tomo 1. Almage&longs;ti &longs;ui novi; fusè quidem lib. 9. <lb/>&longs;ect. 4. cap. 16. num. 24. breviter verò lib. 2. cap. 21. . </s><s>Quibus <lb/>Riccioli ob&longs;ervationibus tantam ego fidem adhibeo, ut licet <lb/>hîc Romæ ea&longs;dem repetere opportuni&longs;&longs;imum mihi foret, de<lb/>mi&longs;&longs;is gravibus per cochleatas Divi Petri in Vaticano &longs;calas <lb/>alti&longs;&longs;imas, & ab omni aëris commotione liberas, fene&longs;tris in<lb/>&longs;uper plurimis ab ima ba&longs;i ad &longs;ummitatem u&longs;que refertas; n<gap/><lb/>luerim tamen me fru&longs;tra fatigare, cùm nec accuratiores ad<lb/>hibere me po&longs;&longs;e putem, nec &longs;i adhibeam, alias præter ip&longs;ius <lb/>leges reperturum me certò &longs;ciam. </s><s>Ex multis igitur de&longs;cen<lb/>&longs;us gravium legibus à Ricciolo locis citatis adductis, duas &longs;e<lb/>quentes affero, quoniam eæ propo&longs;ito no&longs;tro &longs;atisfacient. </s>

<arrow.to.target n="marg220"/><lb/>proponit in Tomo 1. Almage&longs;ti &longs;ui novi; fusè quidem lib. 9. <lb/>&longs;ect. 4. cap. 16. num. 24. breviter verò lib. 2. cap. 21. . </s><s>Quibus <lb/>Riccioli ob&longs;ervationibus tantam ego fidem adhibeo, ut licet <lb/>hîc Romæ ea&longs;dem repetere opportuni&longs;&longs;imum mihi foret, de<lb/>mi&longs;&longs;is gravibus per cochleatas Divi Petri in Vaticano &longs;calas <lb/>alti&longs;&longs;imas, & ab omni aëris commotione liberas, fene&longs;tris in<lb/>&longs;uper plurimis ab ima ba&longs;i ad &longs;ummitatem u&longs;que refertas; no<lb/>luerim tamen me fru&longs;tra fatigare, cùm nec accuratiores ad<lb/>hibere me po&longs;&longs;e putem, nec &longs;i adhibeam, alias præter ip&longs;ius <lb/>leges reperturum me certò &longs;ciam. </s><s>Ex multis igitur de&longs;cen<lb/>&longs;us gravium legibus à Ricciolo locis citatis adductis, duas &longs;e<lb/>quentes affero, quoniam eæ propo&longs;ito no&longs;tro &longs;atisfacient. </s>

<s><margin.target id="marg219"/><emph type="italics"/>Gravium <lb/>naturali <lb/>motu <expan abbr="de-&longs;cendenti&utilde;">de<lb/>&longs;cendentium</expan> <lb/>leges.<emph.end type="italics"/></s>

<s><margin.target id="marg220"/><emph type="italics"/>Riccioli di<lb/>ligentia ex<lb/>acta in ob<lb/>&longs;ervandis le<lb/>gibus gra<lb/>viu<gap/><emph.end type="italics"/></s>

<s><margin.target id="marg220"/><emph type="italics"/>Riccioli di<lb/>ligentia ex<lb/>acta in ob<lb/>&longs;ervandis le<lb/>gibus gra<lb/>vium <expan abbr="de-&longs;cendenti&utilde;">de<lb/>&longs;cendentium</expan><emph.end type="italics"/>.</s>

<s><emph type="italics"/>I. Gravia naturali motu per lineam perpendicularem in aëre de<lb/>

<arrow.to.target n="marg221"/><lb/>&longs;cendentia, moventur difformi velocitate, eâque majore & majore <lb/>ver&longs;us finem motus.<emph.end type="italics"/> </s><s>Lex hæc nota fuit jam inde à viginti & am<lb/>pliùs &longs;æculis in Academijs Phy&longs;icorum, præ&longs;ertim Peripateti<lb/>corum cum Ari&longs;totele lib. 1. de c&oelig;lo cap. 88. inquit Ricciolus, <lb/>patetque oculis, auribus, & tactui; videmus enim gravia de<lb/>&longs;cendentia accelerare magis ac magis motum; audimus vehe<lb/>mentiorem &longs;onum, percipimus validiorem percu&longs;&longs;ionem, ex <lb/>lap&longs;u gravis ac duri è loco altiori, quàm minùs alto. </s>

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<arrow.to.target n="marg222"/><lb/>&longs;cit eo incremento velocitatis, quod e&longs;t inter numeros impares ab uni<lb/>tate numeratos; &longs;eu ita ut &longs;patia æqualibus temporibus tran&longs;mi&longs;&longs;a &longs;int <lb/>inter &longs;e, ut quadrata temporum; &longs;eu ita, ut &longs;patia tran&longs;mi&longs;&longs;a certis <lb/>temporibus habeant inter &longs;e duplicatam proportionem illius, quam ha<lb/>bent tempora quibus &longs;patia illa men&longs;urata fuerint.<emph.end type="italics"/> </s><s>ita Ricciolus lo<lb/>cis citatis, & ante ip&longs;um Galilæus Dialogo 2. de Sy&longs;temate

<pb xlink:href="051/01/153.jpg" pagenum="122"/>Mundi, & Balianus lib 1. de Motu naturali gravium, Propo&longs;it. 6. <lb/>(licet deinde lib. 4. à pag. 110. ad 113. aliter &longs;tatuat) & po&longs;tip&longs;os <lb/>Ga&longs;&longs;endus in Epi&longs;tolis de motu impre&longs;&longs;o à motore, & Kirche<lb/>rus in Mu&longs;urgia lib. 6. parte 1. Lemmate 3. & con&longs;tat experien<lb/>

<arrow.to.target n="marg223"/><lb/>tiâ. </s><s>Numeri pariter impares ab unitate numerati, &longs;unt hi: 1, <lb/>3, 5, 7, 9, 11, 13, 15 &c. </s><s>Itaque &longs;i grave aliquod de&longs;cendens con<lb/>ficit in primo minuto temporis unum &longs;tadium &longs;patij; in &longs;ecundo <lb/>minuto conficiet tria ftadia, in tertio quinque, in quarto &longs;e<lb/>ptem, &c. </s><s>Sic enim &longs;patia æqualibus temporibus confecta, <lb/>erunt inter &longs;e, ut quadrata temporum: nam quadratum mi<lb/>nuti primi e&longs;t 1; & quadratum minuti &longs;ecundi, 4; & quadra<lb/>tum minuti tertij, 9; & quadratum minuti quarti, 16 &c. ut <lb/>apparet inappo&longs;ita tabella. </s><s>

<arrow.to.target n="table1"/><lb/>Cùm igitur grave de&longs;cen<lb/>dens in primo minuto con<lb/>ficiat unum &longs;patium, & in <lb/>&longs;ecundo minuto tria &longs;patia, <lb/>quæ cum uno conficiunt <lb/>quatuor; & in tertio minu<lb/>to quinque &longs;patia, quæ cum <lb/>quatuor efficiunt novem; & in quarto minuto &longs;eptem &longs;patia, <lb/>quæ cum novem efficiunt &longs;exdecim; & in quinto minuto no<lb/>vem &longs;patia, quæ cum &longs;exdecim efficiunt viginti quinque, &c. <lb/>apparet manife&longs;tè, &longs;patia tran&longs;mi&longs;&longs;a à gravi de&longs;cendente, ha<lb/>bere &longs;e, ut quadrata temporum æqualium, quibus percurrun<lb/>tur illa &longs;patia. </s><s>In idem autem recidit, &longs;i dicas, &longs;patia tran&longs;<lb/>mi&longs;&longs;a certis temporibus habere inter &longs;e duplicatam rationem <lb/>&longs;eu proportionem illius proportionis, quam habent tempora, <lb/>quibus &longs;patia illa men&longs;urata fuerint: nam tempus &longs;eu minutum <lb/>&longs;ecundum ad primum, ut apparet ex tabella præcedente, ha<lb/>bet rationem &longs;eu proportionem duplam, &longs;patia verò &longs;ecundo <lb/>minuto confecta, nempe 1 ad 3 (quæ &longs;imul juncta efficiunt 4) <lb/>ad primum minutum habent rationem duplicatam rationis <lb/><gap/>plæ. </s><s>Eadem e&longs;t ratio de cæteris. </s><s>Lege etiam Mundum &longs;u<gap/>-

<arrow.to.target n="table1"/><lb/>Cùm igitur grave de&longs;cen<lb/>dens in primo minuto con<lb/>ficiat unum &longs;patium, & in <lb/>&longs;ecundo minuto tria &longs;patia, <lb/>quæ cum uno conficiunt <lb/>quatuor; & in tertio minu<lb/>to quinque &longs;patia, quæ cum <lb/>quatuor efficiunt novem; & in quarto minuto &longs;eptem &longs;patia, <lb/>quæ cum novem efficiunt &longs;exdecim; & in quinto minuto no<lb/>vem &longs;patia, quæ cum &longs;exdecim efficiunt viginti quinque, &c. <lb/>apparet manife&longs;tè, &longs;patia tran&longs;mi&longs;&longs;a à gravi de&longs;cendente, ha<lb/>bere &longs;e, ut quadrata temporum æqualium, quibus percurrun<lb/>tur illa &longs;patia. </s><s>In idem autem recidit, &longs;i dicas, &longs;patia tran&longs;<lb/>mi&longs;&longs;a certis temporibus habere inter &longs;e duplicatam rationem <lb/>&longs;eu proportionem illius proportionis, quam habent tempora, <lb/>quibus &longs;patia illa men&longs;urata fuerint: nam tempus &longs;eu minutum <lb/>&longs;ecundum ad primum, ut apparet ex tabella præcedente, ha<lb/>bet rationem &longs;eu proportionem duplam, &longs;patia verò &longs;ecundo <lb/>minuto confecta, nempe 1 ad 3 (quæ &longs;imul juncta efficiunt 4) <lb/>ad primum minutum habent rationem duplicatam rationis <lb/>duplæ. </s><s>Eadem e&longs;t ratio de cæteris. </s><s>Lege etiam Mundum &longs;ub-

<pb xlink:href="051/01/154.jpg" pagenum="123"/>terraneum P. Athana&longs;ij Kircheri, ubi in Libro Centro&longs;ophico <lb/>fu&longs;i&longs;&longs;imè pertractat hanc materiam. </s><s>Lege præterea Joannem <lb/>Caramuelem in Cruce Sublimium ingeniorum. </s>

<s><margin.target id="marg222"/>Lex II. gra<lb/>vium <expan abbr="de-&longs;cendenti&utilde;">de<lb/>&longs;cendentium</expan>.</s>

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<s>ESto tubus AB altus uno pede, & alius CD al<lb/>tus quatuor pedibus, &longs;ed uterque æqualium <lb/>foraminum, &longs;ive &longs;emper, &longs;ive non &longs;emper pleni, è <lb/>quibus effluat aqua inæquali velocitate, juxta di<lb/>cta Propo&longs;itione præcedente, Corollario IV. </s><s>Dico, <lb/>velocitates motus & effluxus aquarum e&longs;&longs;e in &longs;ub<lb/>duplicata ratione tuborum, hoc e&longs;t, aquam per <lb/>tubum CD de&longs;cendere, & effluere duplo velo<lb/>ciùs, quàm per tubum AB. </s><s>Quoniam enim aquæ <lb/>

<arrow.to.target n="marg229"/><lb/>motus per tubos æmulatur leges gravium de&longs;cen<lb/>dentium motu naturali, per dicta Propo&longs;it. IX. <lb/>præcedente, hoc e&longs;t, cre&longs;cit in velocitate &longs;ecun-

<arrow.to.target n="fig31"/><lb/>dùm progre&longs;&longs;ionem numerorumim parium ab un<gap/><lb/>tate numeratorum; &longs;i aqua in primo tempore d<gap/><lb/>terminato, v.g. in primo minuto de&longs;cendit à C i<gap/><lb/>E, hoc e&longs;t, ab A in B; in &longs;ecundo minuto d<gap/><lb/>&longs;cendet ab E ad D, ac proinde in fine quarti pa<gap/><lb/>mi mota erit duplo velociùs, quàm in fine prin<gap/><lb/>palmi; Ergo velocitas motus aquæ per tubum CI<gap/><lb/>ad velocitatem aquæ per tubum AB, habet &longs;ul<gap/><lb/>duplicatam rationem altitudinum. </s><s>Eadem e<gap/><lb/>ratio in alijs inæqualibus altitudinibus, quamcu<gap/><lb/>que habeant inter &longs;e proportionem. </s>

<arrow.to.target n="fig31"/><lb/>dùm progre&longs;&longs;ionem numerorumim parium ab uni<lb/>tate numeratorum; &longs;i aqua in primo tempore de<lb/>terminato, v.g. in primo minuto de&longs;cendit à C in <lb/>E, hoc e&longs;t, ab A in B; in &longs;ecundo minuto de<lb/>&longs;cendet ab E ad D, ac proinde in fine quarti pal<lb/>mi mota erit duplo velociùs, quàm in fine primi <lb/>palmi; Ergo velocitas motus aquæ per tubum CD <lb/>ad velocitatem aquæ per tubum AB, habet &longs;ub<lb/>duplicatam rationem altitudinum. </s><s>Eadem e&longs;t <lb/>ratio in alijs inæqualibus altitudinibus, quamcun<lb/>que habeant inter &longs;e proportionem. </s>

<s><margin.target id="marg229"/><emph type="italics"/>Proportio <lb/>velocitatis <lb/>motus aquæ <lb/>de&longs;cenden<lb/>tis per tubos: <lb/><expan abbr="inæquali&utilde;">inæqualium</expan> <lb/>foraminum.<emph.end type="italics"/></s>

<s><emph type="center"/>Pori&longs;ma.<emph.end type="center"/><lb/> </s>

<s>TUbi ergo duplicatam habent rationem illius, quam haben<gap/><lb/>velocitates motus de&longs;cendentis, & effluentis per ip&longs;os aquæ<gap/></s>

<s>TUbi ergo duplicatam habent rationem illius, quam habent <lb/>velocitates motus de&longs;cendentis, & effluentis per ip&longs;os aquæ. </s>

<s><emph type="center"/>Propo&longs;itio XI. Theorema III.<emph.end type="center"/></s>

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<s>QUod a&longs;&longs;eruimus Propo&longs;it. VIII. huius capitis, demon&longs;tran<lb/>;dum hîc e&longs;t, ut fidem ibi obligatam &longs;olvamus. </s>

<s>Ratio igitur ad&longs;ignati hîc & ibi Phænomeni e&longs;t, quòd flu<gap/><lb/>xus aquæ per foramen, &longs;eu quantitas aquæ effluentis, pendet<gap/><lb/>velocitate aquæ eiusdem de&longs;cendentis per tubum, per dict<gap/><lb/>Propo&longs;it. IX. Corollario. I. </s><s>Velocitas autem illa e&longs;t &longs;ubduplicata<gap/><lb/>altitudinum, per dicta Propo&longs;it. X; ergo & quantitas aquæ efflu<lb/>entis ex tubis inæqualium altitudinum &longs;ubduplicata erit <expan abbr="tubor&utilde;">tuborum</expan>. </s>

<s>Ratio igitur ad&longs;ignati hîc & ibi Phænomeni e&longs;t, quòd flu<lb/>xus aquæ per foramen, &longs;eu quantitas aquæ effluentis, pendeti <lb/>velocitate aquæ eiusdem de&longs;cendentis per tubum, per dicta <lb/>Propo&longs;it. IX. Corollario. I. </s><s>Velocitas autem illa e&longs;t &longs;ubduplicata <lb/>altitudinum, per dicta Propo&longs;it. X; ergo & quantitas aquæ efflu<lb/>entis ex tubis inæqualium altitudinum &longs;ubduplicata erit <expan abbr="tubor&utilde;">tuborum</expan>. </s>

<s><emph type="center"/>Propo&longs;itio XII. Theorema IV.<emph.end type="center"/></s>

Line 2818

<s>

<arrow.to.target n="marg232"/><lb/>tudinum, habent rationem duplicatam temporum, quibus <lb/>effluit æqualis aquæ quantitas. </s>

<s><margin.target id="marg232"/><emph type="italics"/>Proportio <lb/>tuborum <gap/><lb/>tempora fl<gap/><lb/>xus aquæ.<emph.end type="italics"/></s>

<s><margin.target id="marg232"/><emph type="italics"/>Proportio <lb/>tuborum ad <lb/>tempora flu<lb/>xus aquæ.<emph.end type="italics"/> </s>

<s><emph type="center"/>Propo&longs;itio XIII. Theorema V.<emph.end type="center"/></s>

Line 2827

<s>ESt Mer&longs;enni in hydraulicis Phænomenis Propo&longs;it. IV. quem <lb/>

<arrow.to.target n="marg233"/><lb/>&longs;equitur P. Andreas Tacquet in &longs;uis hydraulicis Manu&longs;cri<lb/>ptis cap. 4. Phænomen. 7. </s><s>Ratio e&longs;t, quòd licet ex maiori lu<lb/>mine &longs;eu foramine fluat eodem tempore maior aquea columna <lb/>quàm ex minori, tantòque maior exillo quàm ex hoc, quantò <lb/>fuerit maius lumen &longs;eu <expan abbr="foram&etilde;">foramen</expan> illud quàm hoc, quoad aream, per <lb/>dicta Propo&longs;it. 4. huius eapitis; tamen vtraque columna fluit <lb/>ex æquali altitudine tuborum æquâ velocitate phy&longs;icè & ad <lb/>&longs;en&longs;um. </s>

<s><margin.target id="marg233"/><emph type="italics"/>Proportio <gap/><lb/>quæ ad <gap/><lb/><expan abbr="quã">quam</expan> eadem. <lb/>quæ forami<lb/>num in tu<lb/>bu æquè <gap/><lb/>tis.<emph.end type="italics"/></s>

<s><margin.target id="marg233"/><emph type="italics"/>Proportio a<lb/>quæ ad <expan abbr="a-quã">a<lb/>quam</expan> eadem <lb/>quæ forami<lb/>num in tu<lb/>bu æquè al<lb/>tis.<emph.end type="italics"/></s>

<s>Dixi phy&longs;icè & ad &longs;en&longs;um, quia licet gravia eiusdem &longs;pe<lb/>ciei, &longs;ed inæqualis molis ac ponderis, ab eodem ad eundem

<pb xlink:href="051/01/159.jpg" pagenum="128"/>terminum non de&longs;cendant æquè velociter, revera & mathema<lb/>ticè, quemadmodum &longs;entiunt Galilæus Dialogo 2. de Sy&longs;tem. <lb/>Mundi, Ioannes Bapti&longs;ta Balianus lib. 1. de motu naturali gravi<lb/>um &longs;olidorum in Præfat. Nicolaus Cabæus lib. 1. Meteoror. textu <lb/>17. <expan abbr="q.">que</expan> 5. & 6. Arriaga di&longs;put. 4. de Generat. &longs;ect. 5. &longs;ub&longs;ect. 3. <lb/>Ma&longs;trius, Bellutus, & alij (quod ego fal&longs;um exi&longs;timo, mathe<lb/>maticè loquendo, cum Patre Ioanne Bapti&longs;ta Ricciolo, qui <lb/>tom. 1. Almage&longs;ti Novi lib. 2. cap. 21. Propo&longs;it. 2. & lib. 9. &longs;ect. 4. <lb/>num. 24. a&longs;&longs;erit, <emph type="italics"/>duorum gravium eiusdem &longs;peciei & figuræ &longs;ea inæ<lb/>qualis molis ac ponderis, ex eadem altitudine momento eodem dimi&longs;<lb/>&longs;orum, illud naturali motu citiùs de&longs;cendere ad eundem terminum, quod <lb/>e&longs;t gravius;<emph.end type="italics"/> ubi etiam Experimenta multa diver&longs;is annis coram <lb/>multis viris doctis incredibili diligentiâ peracta Bononiæ refert <lb/>num. 13.) tamen in parvis altitudinibus, quales &longs;unt tuborum <lb/>in omni ferè ca&longs;u, tam exigua e&longs;t differentia velocitatum, ut pro <lb/>eâdem &longs;eu æquali cen&longs;eri meritò po&longs;&longs;it. </s>

Line 2857

<s><emph type="italics"/>MEr&longs;ennus in Hydraulicis Phænomenis Propo&longs;it.<emph.end type="italics"/> 8. <emph type="italics"/>ait, con&longs;tare ex <lb/>

<arrow.to.target n="marg235"/><lb/>ob&longs;ervatione, tubum quadrupedalem, cuius ba&longs;is digitalis, uno mi<lb/>nuto temporis totum per lineare lumen exhauriri; tubum verò quadru<lb/>pedalem cuius ba&longs;is pedalis, &longs;patio<emph.end type="italics"/> 144. <emph type="italics"/>minutorum, &longs;eu duabus horis, &<emph.end type="italics"/><lb/>24. <emph type="italics"/>minutis. </s><s>Servatur ergò inter temporaratio ba&longs;ium, vt demon&longs;tra<lb/>vimus: Nam cùm pedis longitudo contineat, ex Mer&longs;enni mente, ut <lb/>vidimus Propo&longs;it. VII. huius Capitis Annot. I. digiti latitudinem<emph.end type="italics"/>

<pb xlink:href="051/01/161.jpg" pagenum="130"/><emph type="italics"/>duodecies; continebit quadratum pedis quadrata digiti<emph.end type="italics"/> 144. <emph type="italics"/>ac proinde <lb/>ba&longs;is pedalis ba&longs;im digitalem centies quadragies quater continebit, <gap/><lb/>diximus in Propo&longs;itione.<emph.end type="italics"/></s>

<pb xlink:href="051/01/161.jpg" pagenum="130"/><emph type="italics"/>duodecies; continebit quadratum pedis quadrata digiti<emph.end type="italics"/> 144. <emph type="italics"/>ac proinde <lb/>ba&longs;is pedalis ba&longs;im digitalem centies quadragies quater continebit, vt <lb/>diximus in Propo&longs;itione.<emph.end type="italics"/></s>

<s><margin.target id="marg235"/>Mer&longs;enni <lb/>ob&longs;ervatio <lb/>in tuborum <lb/>evacuati<lb/>one.</s>

Line 2923

<s>Sit, exempli gratia, altitudo tubi <lb/>ADB pedum 64, luminis &longs;eu foraminis <lb/>diameter DB digitalis, tempus hora una. </s><s><lb/>Oporteat igitur reperire, quantum aquæ <lb/>tribuat, &longs;eu quantam ci&longs;ternam impleat, <lb/>prædictus tubus, intra unam horam. </s><s>So<lb/>lutio pendet ex dictis Propo&longs;it. VII. XIII. <lb/>& VIII. & præterea ex Propo&longs;it. 2. lib. 12. Eu<lb/>clidis. </s><s>Igitur tubum 4 pedes altum, & <lb/>luminis linearis, repræ&longs;entet CPK. </s><s>Ex <lb/>tubo ADB. &longs;ume EDB portionem <expan abbr="etiã">etiam</expan> <lb/>4 pedes altam. </s><s>Et quoniam diameter DB <lb/>digitalis, e&longs;t ad linearem PK, ut 12 ad 1; <lb/>erit per Propo&longs;it. &longs;ecundam lib. 12. Eucli<lb/>dis, lumen DB ad lumen KP, utquadra<lb/>tum diametri DB, ad quadratum diame<lb/>tri PK, hoc e&longs;t, ut 144 ad 1. </s><s>Quare cùm <lb/>per Propo&longs;it. XIII. præcedentem, in tubis <lb/>æquè altis & &longs;emper plenis, &longs;ed inæquali<lb/>um luminum, aqua ab uno effu&longs;a, fit ad a<lb/>quam ab altero effu&longs;am, tempore eodem, <lb/>ut lumen ad lumen; dabit EDB. centies <lb/>quadragies quater plus aquæ, quàm det <lb/>eodem tempore tubus CPK: Sed tubus <lb/>CPK, per Propo&longs;it. VII. præcedentem, <lb/>&longs;patio tredecim &longs;ecundorum temporis dat <lb/>unam libram aquæ: </s><s>Ergo tubus EDB

<arrow.to.target n="fig33"/><lb/>&longs;patio tredecim &longs;ecundorum tomporis da<lb/>bit 144 libras. </s><s><expan abbr="Quoniã">Quoniam</expan> autem in una hora, <lb/>&longs;eu in 60. minutis primis, hoc e&longs;t, in 3600 <lb/>minutis &longs;ecundis, tredecim &longs;ecunda con<lb/>tinentur ducenties &longs;eptuagies &longs;exies, rema<lb/>nentque <gap/>, &longs;ive (facilioris calculi gratia) <lb/>ducenties &longs;eptuagies &longs;epties; &longs;i 144 libras, <lb/>quas tredecim &longs;ecundis dat tubus EDB, <lb/>multiplices per 277, fient ferè 39888. li<lb/>bræ, quas horæ &longs;patio fundit tubus EDB. </s><s><lb/>Inveniatur iam inter 64, altitudinem nem<lb/>pe tubi ADB, & inter 4, altitudinem ni<lb/>mirum portionis EDB, media proporti<lb/>onalis 16. </s><s>Quoniam igitur per Propo&longs;it. <lb/>VIII. præcedentem, aquæ quas tubi ADB, <lb/>& EDB &longs;emper pleni, eodem tempore <lb/>fundunt, &longs;unt in &longs;ubduplicata ratione eius <lb/>quam habent altitudines ADB, & EDB; <lb/>erit aqua quam horæ &longs;patio dat tubus <lb/>EDB, ad aquam quam eodem tempore <lb/>dabit tubus ADB, ut 4 ad 16, &longs;eu ut 1 ad 4. </s><s><lb/>Itaque &longs;i fiat, ut 1 ad 4, ita 39888 (tot enim <lb/>aquæ libras iam o&longs;tendimus horæ &longs;patio <lb/>dare tubum EDP) ad alium numerum, <lb/>nempe ad 159552, quot hic continet vni<lb/>tates, tot aquæ libras horæ &longs;patio dabit tu<lb/>bus ABD &longs;emper plenus, lumen habens <lb/>digitale, altitudinem pedum 64; atque <lb/>adeo tubus prædictus implebit horæ &longs;pa<lb/>tio ci&longs;ternam capacem librarum <lb/>aquæ 159552. </s>

<arrow.to.target n="fig33"/><lb/>&longs;patio tredecim &longs;ecundorum tomporis da<lb/>bit 144 libras. </s><s><expan abbr="Quoniã">Quoniam</expan> autem in una hora, <lb/>&longs;eu in 60. minutis primis, hoc e&longs;t, in 3600 <lb/>minutis &longs;ecundis, tredecim &longs;ecunda con<lb/>tinentur ducenties &longs;eptuagies &longs;exies, rema<lb/>nentque 12/<gap/>, &longs;ive (facilioris calculi gratia) <lb/>ducenties &longs;eptuagies &longs;epties; &longs;i 144 libras, <lb/>quas tredecim &longs;ecundis dat tubus EDB, <lb/>multiplices per 277, fient ferè 39888. li<lb/>bræ, quas horæ &longs;patio fundit tubus EDB. </s><s><lb/>Inveniatur iam inter 64, altitudinem nem<lb/>pe tubi ADB, & inter 4, altitudinem ni<lb/>mirum portionis EDB, media proporti<lb/>onalis 16. </s><s>Quoniam igitur per Propo&longs;it. <lb/>VIII. præcedentem, aquæ quas tubi ADB, <lb/>& EDB &longs;emper pleni, eodem tempore <lb/>fundunt, &longs;unt in &longs;ubduplicata ratione eius <lb/>quam habent altitudines ADB, & EDB; <lb/>erit aqua quam horæ &longs;patio dat tubus <lb/>EDB, ad aquam quam eodem tempore <lb/>dabit tubus ADB, ut 4 ad 16, &longs;eu ut 1 ad 4. </s><s><lb/>Itaque &longs;i fiat, ut 1 ad 4, ita 39888 (tot enim <lb/>aquæ libras iam o&longs;tendimus horæ &longs;patio <lb/>dare tubum EDP) ad alium numerum, <lb/>nempe ad 159552, quot hic continet vni<lb/>tates, tot aquæ libras horæ &longs;patio dabit tu<lb/>bus ABD &longs;emper plenus, lumen habens <lb/>digitale, altitudinem pedum 64; atque <lb/>adeo tubus prædictus implebit horæ &longs;pa<lb/>tio ci&longs;ternam capacem librarum <lb/>aquæ 159552. </s>

Line 2950

<s>Revocentur primò 3000 pedes cubici aquæ ad heminas <lb/>

<arrow.to.target n="fig34"/><lb/>&longs;eu libras (quod fiet, &longs;i 3000 multiplica<lb/>bis per 72. fiatq, numerus librarum 216000, <lb/>(quandoquidem unus pes cubicus conficit <lb/>72 libras, ut diximus Propo&longs;it: præceden<lb/>te in Annotat:) Deinde &longs;umatur ex ADB, <lb/>tubus quadrupedalis EDB: Demum inter <lb/>ADB altitudinem, id e&longs;t, 16, & EDB alti<lb/>tudinem, id e&longs;t, 4, inveniatur medius nu<lb/>merus proportionalis 8. </s><s>Igitur, per Pro<lb/>po&longs;it. VIII. præcedentem, aqua fluens per <lb/>tubum EDB, e&longs;t ad aquam per tubum <lb/>ADB eodem tempore fluentem, ut 4 ad 8, <lb/>in ratione videlicet &longs;ubduplicata altitudi<lb/>num EDB 4 pedum, & ADB 16 pedum. </s><s><lb/>Quo ergo tempore dat tubus ADB 3000 <lb/>pedes cubicos aquæ, id e&longs;t, 216000 hemi<lb/>nas &longs;eu libras; eodem tempore EDB dabit duplò minus, &longs;eu

<arrow.to.target n="fig35"/><lb/>heminas &longs;olùm 108000. </s><s>A&longs;&longs;umatur jam, <lb/>ut antea, tubus quadrupedalis CPK lu<lb/>minis linearis. </s><s>Tubi EDB, CPK, &longs;unt æ<lb/>què alti; & lumina DB, PK, dantur, per <gap/><lb/>Propo&longs;it. lib. 12. Euclid. nempe 1, & 144 <lb/>lineæ quadratæ; aqua etiam quam CPK <lb/>fundit &longs;patio 13 &longs;ecundorum, datur, per VII. <lb/>præcedentem, una nimirum libra: Qua<lb/>re cùm quantitates aquæ à tubis æquè al<lb/>tis eodem tempore effu&longs;æ &longs;int ut lumina, <lb/>per Propo&longs;it. XIII. præcedentem, &longs;i fiat ut <lb/>lumen PK, ad lumen DB, id e&longs;t, ut 1 ad <lb/>144, ita aqua unius ad aquam alterius; pro<lb/>ducetur numerus heminarum &longs;eu libra<lb/>rum, quas effundit tubus EDB &longs;patio tre<lb/>decim &longs;ecundorum, nempe 144. </s><s>Quo<lb/>niam igitur tubus EDB 144 libras dat &longs;patio tredecim &longs;ecundo<lb/>rum; libras 108000 quanto tempore dabit? </s><s>Fiat, ut 144 libræ <lb/>ad <gap/> hoc e&longs;t, ad tredecim &longs;ecunda, ita 108000 ad 9750; dabit <lb/>hic numerus &longs;ecunda, quibus tubus EDB dat libras 108000. </s><s><lb/>Sed o&longs;ten&longs;um e&longs;t &longs;uprà, quo tempote tubus EDB dat 108000 <lb/>libras, tubum ADB dare libras 216000, id e&longs;t, 3000 pedes cu<lb/>bicos aquæ; Inventum e&longs;t igitur tempus, quo tubus ADB &longs;em<lb/>per plenus fundit datam aquam 3000 pedum cubicorum, nem<lb/>pe 9750 &longs;ecunda, &longs;eu 162 prima; quæ faciunt horas 2, & 42 <lb/>minuta prima. </s>

<arrow.to.target n="fig35"/><lb/>heminas &longs;olùm 108000. </s><s>A&longs;&longs;umatur jam, <lb/>ut antea, tubus quadrupedalis CPK lu<lb/>minis linearis. </s><s>Tubi EDB, CPK, &longs;unt æ<lb/>què alti; & lumina DB, PK, dantur, per 1. <lb/>Propo&longs;it. lib. 12. Euclid. nempe 1, & 144 <lb/>lineæ quadratæ; aqua etiam quam CPK <lb/>fundit &longs;patio 13 &longs;ecundorum, datur, per VII. <lb/>præcedentem, una nimirum libra: Qua<lb/>re cùm quantitates aquæ à tubis æquè al<lb/>tis eodem tempore effu&longs;æ &longs;int ut lumina, <lb/>per Propo&longs;it. XIII. præcedentem, &longs;i fiat ut <lb/>lumen PK, ad lumen DB, id e&longs;t, ut 1 ad <lb/>144, ita aqua unius ad aquam alterius; pro<lb/>ducetur numerus heminarum &longs;eu libra<lb/>rum, quas effundit tubus EDB &longs;patio tre<lb/>decim &longs;ecundorum, nempe 144. </s><s>Quo<lb/>niam igitur tubus EDB 144 libras dat &longs;patio tredecim &longs;ecundo<lb/>rum; libras 108000 quanto tempore dabit? </s><s>Fiat, ut 144 libræ <lb/>ad <gap/> hoc e&longs;t, ad tredecim &longs;ecunda, ita 108000 ad 9750; dabit <lb/>hic numerus &longs;ecunda, quibus tubus EDB dat libras 108000. </s><s><lb/>Sed o&longs;ten&longs;um e&longs;t &longs;uprà, quo tempote tubus EDB dat 108000 <lb/>libras, tubum ADB dare libras 216000, id e&longs;t, 3000 pedes cu<lb/>bicos aquæ; Inventum e&longs;t igitur tempus, quo tubus ADB &longs;em<lb/>per plenus fundit datam aquam 3000 pedum cubicorum, nem<lb/>pe 9750 &longs;ecunda, &longs;eu 162 prima; quæ faciunt horas 2, & 42 <lb/>minuta prima. </s>

Line 3017

<s><margin.target id="marg245"/><emph type="italics"/><expan abbr="Altitudin&etilde;">Altitudinem</expan> <lb/>Scaturigi<lb/>nis fontis in<lb/>venire per <lb/>tubos fluen<lb/>tis.<emph.end type="italics"/></s>

<s>FIat notum lumen, per quod fontis a<lb/>qua fluat; aut fonti lumen notæ ma<lb/>gnitudinis applica, v.g. lineare. </s><s>Ob<lb/>&longs;erva deinde quot aquæ libras fons per <lb/>lineare <expan abbr="lum&etilde;">lumen</expan> effundat &longs;patio unius minuti primi, &longs;eu 60 minuto<lb/>rum &longs;ecundorum; &longs;itque numerus ille librarum B. </s><s>Quoniam <lb/>igitur per Propo&longs;it. VII. huius capitis, tubus qua drupedalis &longs;emper <lb/>plenus per lumen lineare effundit &longs;patio tiedecim &longs;ecundorum <lb/>vnam libram, & con&longs;equenter &longs;patio 60 &longs;ecundorum, &longs;eu unius <lb/>minuti primi, libras 4<gap/>: & præterea, quoniam per Pori&longs;ma I. <lb/>Propo&longs;it. VIII. huius capitis, altitudines tuborum, habentium <lb/>idem &longs;eu æquale lumen, &longs;unt in duplicata ratione eius quam ha<lb/>bent aquæ quantitates per tubos <expan abbr="eod&etilde;">eodem</expan> tempore effusæ: &longs;i fiat, ut <lb/>4<gap/> lib. ad numerum librarum B, ita altitudo 4 pedum, ad aliud, <lb/>nempe ad altitudinem numeri M; & iterum, ut 4 ad M, ita M <lb/>ad N; dabit numerus N altitudinem &longs;caturiginis in pedibus, <lb/>eò quòd ratio 4 ad N &longs;it duplicata rationis 4 ad M, &longs;eu ratic<lb/>nis 4<gap/> ad B, nimirum aquæ ad aquam. </s>

<s>FIat notum lumen, per quod fontis a<lb/>qua fluat; aut fonti lumen notæ ma<lb/>gnitudinis applica, v.g. lineare. </s><s>Ob<lb/>&longs;erva deinde quot aquæ libras fons per <lb/>lineare <expan abbr="lum&etilde;">lumen</expan> effundat &longs;patio unius minuti primi, &longs;eu 60 minuto<lb/>rum &longs;ecundorum; &longs;itque numerus ille librarum B. </s><s>Quoniam <lb/>igitur per Propo&longs;it. VII. huius capitis, tubus quadrupedalis &longs;emper <lb/>plenus per lumen lineare effundit &longs;patio tredecim &longs;ecundorum <lb/>vnam libram, & con&longs;equenter &longs;patio 60 &longs;ecundorum, &longs;eu unius <lb/>minuti primi, libras 4 8/13: & præterea, quoniam per Pori&longs;ma I. <lb/>Propo&longs;it. VIII. huius capitis, altitudines tuborum, habentium <lb/>idem &longs;eu æquale lumen, &longs;unt in duplicata ratione eius quam ha<lb/>bent aquæ quantitates per tubos <expan abbr="eod&etilde;">eodem</expan> tempore effusæ: &longs;i fiat, ut <lb/>4 8/13 lib. ad numerum librarum B, ita altitudo 4 pedum, ad aliud, <lb/>nempe ad altitudinem numeri M; & iterum, ut 4 ad M, ita M <lb/>ad N; dabit numerus N altitudinem &longs;caturiginis in pedibus, <lb/>eò quòd ratio 4 ad N &longs;it duplicata rationis 4 ad M, &longs;eu ratic<lb/>nis 4 8/13 ad B, nimirum aquæ ad aquam. </s>

<s><emph type="center"/>Propo&longs;itio XXIII. Problema VII.<emph.end type="center"/></s>

Line 3129

<arrow.to.target n="marg253"/><lb/>line is propo&longs;itis. </s><s>Itaque &longs;ipropo&longs;itis duobus tubis inveniendus &longs;it <lb/>vel medius, veltertius proportionalis; coniunge lineas rectas tubis da<lb/>tis æquales; & operare ut dictum, & invenies quod quæris. </s><s>Quòd &longs;i <lb/>tubi propo&longs;iti, ac lineæ ip&longs;is æquales nimis e&longs;&longs;ent longæ, ac proinde minùs <lb/>commodè circulo includi po&longs;&longs;ent; accipe ip&longs;arum &longs;ubmultiplices, v.g. <lb/>dimidiam, tertiam, quartam, &c. partem, & cum ip&longs;is procede <lb/>ut dictum; eritque inventa linea æquè &longs;ubmultiplex <lb/>lineæ aut tubi quæ&longs;iti.<emph.end type="italics"/></s>

<s><margin.target id="marg253"/>Tubum me<lb/>dium <gap/> t<gap/><lb/>tium p<gap/>r<lb/>tionalem in<lb/>venire.</s>

<s><margin.target id="marg253"/>Tubum me<lb/>dium & ter<lb/>tium propor<lb/>tionalem in<lb/>venire.</s>

<s><emph type="center"/>Propo&longs;itio V.<emph.end type="center"/></s>

Line 3267

<s><emph type="italics"/>QVanta verò &longs;it &longs;alientium in quavis tubi &longs;upra horizontem eleva<lb/>tione longitudo, inve&longs;tigandum e&longs;t ob&longs;ervatione, quam quilibet <lb/>facilè poterit facere modo dicto Propo&longs;itione præcedente. </s><s>Mer&longs;ennus <lb/>in &longs;uis Phænomenis Propo&longs;it.<emph.end type="italics"/> XVI. <emph type="italics"/>proponit paucas ex multis in &longs;equen<lb/>ti tabella, in quâ Prima Columna continet varias tubi pedalis, velpo<lb/>tiùs luminis tubi, &longs;upra horizontem elevationes in pedibus, ita ut prima <lb/>elevatio &longs;it unius pedis, &longs;ecunda duorum pedum, &c: Secunda verò co<lb/>lumna exhibet &longs;alientium longitudines in pedibus & digitis. </s><s>Addidi<emph.end type="italics"/>

<arrow.to.target n="marg266"/><lb/><emph type="italics"/>ego ex eodem Mer&longs;enno Propo&longs;it.<emph.end type="italics"/> 29. <emph type="italics"/>&longs;alientem tubi pedalis elevati p<gap/><lb/>uno cum dimidio &longs;upra horizontem.<emph.end type="italics"/></s>

<arrow.to.target n="marg266"/><lb/><emph type="italics"/>ego ex eodem Mer&longs;enno Propo&longs;it.<emph.end type="italics"/> 29. <emph type="italics"/>&longs;alientem tubi pedalis elevati pede <lb/>uno cum dimidio &longs;upra horizontem.<emph.end type="italics"/></s>

<s><margin.target id="marg266"/><emph type="italics"/>Salientium <lb/>tabella ex <lb/>Mer&longs;enno.<emph.end type="italics"/></s>

Line 3411

<s><emph type="center"/>De fluxu Aquæ per diver&longs;a eiu&longs;dem va&longs;is <lb/>aut tubi foramina.<emph.end type="center"/><lb/>

<arrow.to.target n="marg275"/></s>

<s><margin.target id="marg275"/><emph type="italics"/>Proprieta<lb/>tes <gap/>uxus <lb/>aqu<gap/> per di<lb/>ver&longs;a eius<lb/>dem tubs <lb/>au: va&longs;is <lb/>foramma.<emph.end type="italics"/></s>

<s><margin.target id="marg275"/><emph type="italics"/>Proprieta<lb/>tes fluxus <lb/>aquæ per di<lb/>ver&longs;a eius<lb/>dem tubi <lb/>aut va&longs;is <lb/>foramina.<emph.end type="italics"/></s>

<s>HActenus locuti &longs;umus de aquæ fluxu per foramina tu<lb/>borum <expan abbr="verticali&utilde;">verticalium</expan> tum horizontalia, tum lateralia, & <lb/>tubos cum tubis, aquas cum aquis, foramina cum fo<lb/>raminibus diver&longs;orum tuborum comparavimus; <lb/>nunc eiu&longs;dem tubi & va&longs;is <expan abbr="cuiu&longs;cunq;">cuiu&longs;cunque</expan> foramina diver&longs;a inter &longs;e&longs;e

<pb xlink:href="051/01/190.jpg" pagenum="159"/>conferemus breviter & methodicè, in&longs;i&longs;tentes ve&longs;tigijs Ioannis <lb/>Bapti&longs;tæ Baliani lib. 6. de Motu naturali gravium & liquidorum. </s><s><lb/>Quibus tamen multa addi po&longs;&longs;unt ex dictis Capite præcedenti. </s><s><lb/>Sermo autem e&longs;t hîc etiam, uti in præcedentibus, de tu<lb/>bis ac va&longs;is verticaliter erectis, &longs;ive cylindrica illa &longs;int, &longs;ive pri&longs;<lb/>matica; & &longs;ive foramina &longs;int rectangula, &longs;ive circularia. </s><s>Præmit<lb/>tamus ergo &longs;equentes. </s>

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